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# Lecture9-new - 6.254 : Game Theory with Engineering...

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6.254 : Game Theory with Engineering Applications Lecture 9: Computation of NE in ﬁnite games Asu Ozdaglar MIT March 4, 2010 1

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Game Theory: Lecture 9 Introduction Introduction In this lecture, we study various approaches for the computation of mixed Nash equilibrium for ﬁnite games. Our focus will mainly be on two player ﬁnite games (i.e., bimatrix games). We will also mention extensions to games with multiple players and continuous strategy spaces at the end. The two survey papers [von Stengel 02] and [McKelvey and McLennan 96] provide good references for this topic. 2
Game Theory: Lecture 9 Zero-Sum Finite Games Zero-Sum Finite Games We consider a zero-sum game where we have two players. Assume that player 1 has n actions and player 2 has m actions. We denote the n × m payoﬀ matrices of player 1 and 2 by A and B . Let x denote the mixed strategy of player 1, i.e., x X , where X = { x | n i = 1 x i = 1, x i 0 } , and y denote the mixed strategy of player 2, i.e., y Y , where Y = { y | m j = 1 y j = 1, y j 0 } . Given a mixed strategy proﬁle ( x , y ) , the payoﬀs of player 1 and player 2 can be expressed in terms of the payoﬀ matrices as, u 1 ( x , y ) = x T Ay , u 2 ( x , y ) = x T By . 3

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Game Theory: Lecture 9 Zero-Sum Finite Games Zero-Sum Finite Games Recall the deﬁnition of a Nash equilibrium: A mixed strategy proﬁle ( x * , y * ) is a mixed strategy Nash equilibrium if and only if ( x * ) T Ay * x T Ay * , for all x X , ( x * ) T By * ( x * ) T By , for all y Y . For zero-sum games, we have B = - A , hence the preceding relation becomes ( x * ) T Ay * ( x * ) T Ay , for all y Y . Combining the preceding, we obtain x T Ay * ( x * ) T Ay * ( x * ) T AY , for all x X , y Y , i.e., ( x * , y * ) is a saddle point of the function x T Ay deﬁned over X × Y . Note that a vector ( x * , y * ) is a saddle point if x * X , y * Y , and sup x X x T Ay * = ( x * ) T Ay * = inf y Y ( x * ) T Ay . (1) 4
Game Theory: Lecture 9 Zero-Sum Finite Games Zero-Sum Finite Games For any function φ : X × Y 7→ R , we have the minimax inequality : sup x X inf y Y φ ( x , y ) inf y Y sup x X φ ( x , y ) , (2) Proof: To see this, for every ¯ x X , write inf y Y φ ( ¯ x , y ) inf y Y sup x X φ ( x , y ) and take the supremum over ¯ x X of the left-hand side. Eq. (1) implies that inf y Y sup x X x T Ay sup x X x T Ay * = ( x * ) T Ay * = inf y Y ( x * ) T Ay sup x X inf y Y x T Ay , which combined with the minimax inequality [cf. Eq. (2)], proves that equality holds throughout in the preceding. Hence, a mixed strategy proﬁle ( x * , y * ) is a Nash equilibrium if and only if ( x * ) T Ay * = inf y Y sup x X x T Ay = sup x X inf y Y x T Ay . We refer to ( x * ) T Ay * as the value of the game . 5

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Game Theory: Lecture 9 Zero-Sum Finite Games Zero-Sum Finite Games We next show that ﬁnding the mixed strategy Nash equilibrium strategies and the value of the game can be written as a pair of linear optimization problems.
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## This note was uploaded on 05/08/2010 for the course CS 6.254 taught by Professor Asuozdaglar during the Spring '10 term at MIT.

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Lecture9-new - 6.254 : Game Theory with Engineering...

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